Erratum to: Comprehensive Bayesian analysis of rare (semi)leptonic and radiative B decays

1 Erratum to: Eur Phys J C (2014) 74:2897 DOI 10.1140/epjc/s10052-014-2897-0

When revisiting our fits in order to expand the above work, errors in the implementation of analytical expressions of the observables have been encountered. One of these errors affects the branching ratio of $B\rightarrow K\ell ^+\ell ^-$ at low $q^2$ in the presence of chirality-flipped operators. Carefully checking our results, we also found that systematic uncertainties of the lattice results of the $B\rightarrow K^{(*)}$ form factors had been incorrectly neglected.

After correcting these errors, we replace Tables 2 (“Postdiction” rows only), 3, 6, 4, and 5, as well as Figs. 2, 3, and 4. We also replace selected parts of Sect. 4 that are not given in the tables.

Fig. 2

Credibility regions of the Wilson coefficients $\mathcal {C}_{7,9,10}^{\mathrm {}}$ obtained from the fit of the “full” data set after the EPSHEP 2013 conference at $68.3\,\%$, $95.4\,\%$, $99.7\,\%$ (dark, normal, and light red) probability. The SM-like solution $A$ (upper row) and the flipped-sign solution $B$ (lower row) are magnified. Overlaid are the results of the fit to the “selection” data set at $68.3\,\%$, $95.4\,\%$, $99.7\,\%$ (solid, dashed, and dash-dotted blue lines) probability. The projection of the SM point is represented by the black diamond, whereas the black and blue crosses mark the best-fit points

Fig. 3

Credibility regions obtained from the fit in the SM+SM$'$  model. We show the results of the “full” data set after the EPSHEP 2013 conference at $68.3\,\%$ (dark red) and $95.4\,\%$ (light red) probability. The regions from the fit of the “full (+FF)” data set are overlaid by blue lines at $68.3\,\%$ (solid) and $95.4\,\%$ (dashed) probability. The projection of the SM point is shown by the black diamond and the black and blue crosses mark the best-fit point in the respective 2D plane

Fig. 4

Credibility regions obtained from the fit in the SM+SM$'$ $(9)$ model. We show the results of the “full” data set at $68.3\,\%$ (dark red) and $95.4\,\%$ (red) probability. The regions from the fit of the “full (+FF)” data set are overlaid by blue lines at $68.3\,\%$ (solid) and $95.4\,\%$ (dashed) probability. The projection of the SM point is represented by the black diamond, whereas the black and blue crosses mark the best-fit points

Table 2 Postdictions for the optimized observables $P'_{4,5,6}$ in various $q^2$ bins. Note that the uncertainties correspond to $68\,\%$ credibility intervals that arise from variation of all fit parameters. Note further that the postdiction of $\langle P'_5\rangle _{[1,6]}$ in SM+SM$'$ consists of two distinct regions around $-0.3$ and $-0.4$ from solutions $(A' + B')$ and $(C' + D')$, respectively. $\dagger$: Values have been adjusted to match the theory convention for the observable

Table 3 Goodness of fit and posterior evidence (ratio) for various combinations of constraints and fit models. Individual solutions are labeled as $A$ and $B$ in the SM($\nu -$only) and the SM, and $A'$ through $D'$ in the SM+SM$'$ and $A'$ in the SM+SM$'$ $(9)$. The solutions with SM-like and flipped signs of $\mathcal {C}_{i}^{\mathrm {}}$ are $A^{(\prime )}$ and $B^{(\prime )}$, respectively. For the definitions of $P(D|M)$ and $R$, see Eq. (4.1) in the original article

Table 4 The 68- and 95 %-credibility intervals and the local modes of the marginalized 1D posterior distributions of the Wilson coefficients at $\mu = 4.2$ GeV, $P(\mathcal {C}_{i}^{\mathrm {}}|D),\, i = 7,9,10,7',9',10'$, for nominal priors of nuisance parameters in the various scenarios. Note that for the SM+SM$'$-scenario the individual solutions cannot be disentangled within the 1D posterior distributions, unlike for the SM-scenario. For comparison, the SM values of the Wilson coefficients read $\mathcal {C}_{7}^{\mathrm {SM}} = -0.34$, $\mathcal {C}_{9}^{\mathrm {SM}} = +4.27$, $\mathcal {C}_{10}^{\mathrm {SM}} = -4.17$, $\mathcal {C}_{7'}^{\mathrm {SM}} = -0.01$, $\mathcal {C}_{9'}^{\mathrm {SM}} = \mathcal {C}_{10'}^{\mathrm {SM}} = 0$

Table 5 Compilation of the pull values in units of $\sigma$ at the SM-like best-fit points $A$ in the SM  fit (left columns) and $A'$ in the SM+SM$'$  fit (right columns), listed per experiment and observable. Only pull values for fits with the “full” data set are listed. The single CLEO measurement of $\mathcal {B}(B\rightarrow K^*\gamma )$ has a pull value $+0.3\sigma$ in both the SM  and the SM+SM$'$  fits. The pull values for the SM($\nu$-only) fit deviate by less than $0.3\sigma$ from those of the SM  fit for the “full” data set, with the exception of $+2.3\sigma$ for the LHCb measurement of $\langle P'_5\rangle _{[1,6]}$. The pull values in scenario SM+SM$'$  with the “full (+FF)” data set deviate by less than $0.4\sigma$ from those given for SM+SM$'$  “full”, except for the LHCb measurements of $\langle A_T^{(2)} \rangle _{[1,6]}$ with $+1.1\sigma$, $\langle A_T^{(\mathrm{re})}\rangle _{[1,6]}$ with $+0.6\sigma$ and $\langle P'_5\rangle _{[1,6]}$ with $+1.2\sigma$. The pull values for SM  “full (+FF)” deviate by less than $0.3\sigma$ from those of SM  “full” except for $\langle P'_5\rangle _{[1,6]}$ with slightly increased by $+0.4\sigma$

Table 6 1D-marginalized posterior results at 68 % probability in comparison to the prior inputs for the various $B\rightarrow K^*$ (upper rows) and $B\rightarrow K$ (middle two rows) form-factor parameters. The results are shown for the “full” (left) and “full (+FF)” (right) data set in various scenarios. The priors for the “full” data set comprise LCSR [2] inputs combined with the additional constraints Eqs. (6.1)–(6.3); see the original article, and $B\rightarrow K$ lattice results [3], whereas for “full (+FF)” the $B\rightarrow K^*$ lattice results [4] are added. Note that the marginalization has been performed over all solutions $A,B$ in the case of SM  and $A'-D'$ in the case of SM+SM$'$

While our main conclusions stay as they are, some details are adjusted. Our revised conclusions are given at the end of this erratum.

2 Result

2.1 Statistical approach

For the “selection” data set, we erroneously included the observables $P_i^\prime$ at high $q^2$. Hence there are now $N=20$ experimental inputs, two theory constraints, and $\dim \mathbf {\nu }= 24$.

2.2 Fit in the SM basis

Eq. (4.7) correctly reads

$$\begin{aligned} \varDelta _{9} = \mathcal {C}_{9}^{\mathrm {}} - \mathcal {C}_{9}^{\mathrm {SM}} \simeq -1.7 \pm 0.7. \end{aligned}$$

(4.7)

Our corrected result for the deviation in the ($\mathcal {C}_{7}^{\mathrm {}}$ – $\mathcal {C}_{9}^{\mathrm {}}$) from the SM expectation is $\simeq 2.5\sigma$, and solution $A$ is favored over solution $B$ with $R_{A}{:}R_{B} = 82\,\%{:}18\,\%$. Solution $A$ is described by the 1D marginalized $68\,\%$ credibility regions

$$\begin{aligned}&\varDelta _7 = 0 \pm 0.02, \quad \varDelta _9 = -0.5 \pm 0.3,\quad \varDelta _{10} = -0.2 \pm 0.3, \end{aligned}$$

and loses the model comparison with the SM($\nu$-only) model with the corrected Bayes factor of

$$\begin{aligned} \frac{P(\text {full} | \text {SM})}{P(\text {full | SM}(\nu \text {-}\mathrm{only}))} \Big |_{A} = 1{:}48. \end{aligned}$$

(4.8)

Including the ${B \rightarrow K^{*}}$ form factors from the lattice, we now find

$$\begin{aligned} \varDelta _7 \!=\! 0.0 \pm 0.02,\quad \varDelta _9 \!=\! -0.5 \pm 0.3 ,\quad \varDelta _{10} \!=\! -0.1 \pm 0.3, \end{aligned}$$

and

$$\begin{aligned} \frac{P(\text {full (+FF)} | \text {SM})}{P(\text {full (+FF)} | \mathrm{SM}(\nu \text {-}\mathrm{only}))} \Big |_{A} = 1{:}43. \end{aligned}$$

For the data set “selection”, the credibility regions in Fig. 2 are larger now as the observables $P_i^\prime$ at high $q^2$ are no longer part of it.

2.3 Fit in the extended SM+SM$'$  basis

We now find that, of all four solutions, $A'$ and $D'$ dominate over $B'$ and $C'$ in terms of the posterior mass:

$$\begin{aligned} R_{A'}{:}R_{B'}{:}R_{C'}{:}R_{D'} = 37\,\%{:}14\,\%{:}15\,\%{:}34\,\%. \end{aligned}$$

The SM-like solution $A'$ exhibits a good fit, with a $p$ value of $0.07$. We find agreement between $A'$ and the SM point at $\sim 1\sigma$. The $68\,\%$ probability regions are

$$\begin{aligned} \varDelta _{7} \!=\! 0.01 \pm 0.02, \quad \varDelta _{9} \!=\! -0.8 \pm 0.4, \quad \varDelta _{10} \!=\! -0.2 \pm 0.3. \end{aligned}$$

For further goodness-of-fit criteria for the other solutions we refer to the revised Table 3. The model comparison now yields a Bayes factor of

$$\begin{aligned} \frac{P(\text {full} | \text {SM+SM'})}{P(\text {full} | \mathrm{SM}(\nu \text {-}\mathrm {only}))}\Big |_{A'} = 1{:}401. \end{aligned}$$

(4.9)

For the “full (+FF)” data set, i.e. including lattice results of ${B \rightarrow K^{*}}$ form factors, we now find

$$\begin{aligned} \varDelta _{7} = 0.00 ^{+0.03}_{-0.02},\qquad \varDelta _{9} = -0.8 ^{+0.4}_{-0.3}, \qquad \varDelta _{10} = -0.1 \pm 0.3, \end{aligned}$$

and

$$\begin{aligned} \frac{P(\text {full (+FF)} | \text {SM+SM'})}{P(\text {full (+FF)} | \mathrm{SM}(\nu \text {-}\mathrm {only}))}\Big |_{A'} = 1{:}148. \end{aligned}$$

(4.10)

In the SM+SM$'$ $(9)$-scenario, the SM-like solution $A'$ for the fit now reads

$$\begin{aligned} \varDelta _{9} = -0.8^{+0.4}_{-0.3}, \qquad \varDelta _{9'} = +0.5 \pm 0.6, \end{aligned}$$

which remains compatible with the findings of [1]. Our best-fit point is compatible with the SM point at less than $1\sigma$. For solution $A'$, the Bayes factor reads

$$\begin{aligned} \frac{P(\text {full} | \text {SM+SM' (9)})}{P(\text {full} | \mathrm{SM}(\nu \text {-}\mathrm {only}))}\Big |_{A'} = 1{:}3, \end{aligned}$$

(4.11)

which is slightly in favor of the SM. However, adding the lattice QCD results on the $B\rightarrow K^*$ form factors evens the odds:

$$\begin{aligned} \frac{P(\text {full (+FF)} | \text {SM+SM' (9)})}{P(\text {full (+FF)} | \mathrm{SM}(\nu \text {-}\mathrm {only}) )}\Big |_{A'} = 1{:}1 . \end{aligned}$$

(4.12)

3 Conclusions

Our Bayesian analysis indicates that the standard model provides an adequate description of the available measurements of rare leptonic, semileptonic, and radiative $B$ decays. Compared to our previous analysis [5], we determine the Wilson coefficients $\mathcal {C}_{7,9,10}^{\mathrm {}}$ more accurately, dominantly due to the reduction of the experimental uncertainties in the exclusive decays and the addition of the inclusive decay $B\rightarrow X_s\gamma$.

Contrary to all similar analyses, our fits include the theory uncertainties explicitly through nuisance parameters. We observe that tensions in the angular and optimized observables in $B\rightarrow K^*\ell ^+\ell ^-$ decays can be lifted through $(10$–$20)\,\%$ shifts in the transversity amplitudes at large recoil due to subleading contributions. These shifts are present within the SM as well as the model-independent extension of real-valued Wilson coefficients $\mathcal {C}_{7,9,10}^{\mathrm {}}$. For the scenarios introducing additional chirality-flipped coefficients $\mathcal {C}_{7',9',10'}^{\mathrm {}}$, the shifts reduce to a few percent (except $\zeta _{K^*}^{L\perp }$). We find $|\mathcal {C}_{9',10'}^{\mathrm {}}| \lesssim 5$ at $95\,\%$ probability; see Fig. 3 and Table 4, for the right-handed couplings, which holds in the absence of scalar and tensor contributions. These constraints are insensitive to the shape (Gaussian vs. flat) of the priors of subleading corrections.

Among the information inferred from the data are constraints on the parameters of the $B\rightarrow K^{(*)}$ form factors. We have performed all fits with and without the very recent lattice $B\rightarrow K^*$ form factor predictions [4]. In both cases, the posterior ranges of the Wilson coefficients $\mathcal {C}_{7,7',10,10'}^{\mathrm {}}$ are essentially the same apart from minor shifts in $\mathcal {C}_{9,9'}^{\mathrm {}}$. Again in both cases, the posteriors of the $B\rightarrow K^{*}$ form-factor parameters are very similar. This comes as a surprise given the large difference in prior uncertainties but implies that the combination of measurements supports the lattice input, even independently of the scenario.

The rough picture emerging from the current data may be summarized as follows. The low-$q^2$ $B\rightarrow K^*\ell ^+\ell ^-$ data prefer a negative new-physics contribution to $\mathcal {C}_{9}^{\mathrm {}}$ [6], which is not supported by $B\rightarrow K \ell ^+\ell ^-$ data unless one allows a positive contribution to $\mathcal {C}_{9'}^{\mathrm {}}$ (or alternatively $\mathcal {C}_{10'}^{\mathrm {}}$) [1]. Our Bayesian analysis shows strong support for the standard model SM($\nu$-only) compared to additional new physics in Wilson coefficients $\mathcal {C}_{7,9,10}^{\mathrm {}}$ in the SM-scenario and/or chirality-flipped $\mathcal {C}_{7',9',10'}^{\mathrm {}}$ in the SM+SM$'$-scenario in terms of Bayes factors. Only a reduced scenario SM+SM$'$ $(9)$ of the two Wilson coefficients $\mathcal {C}_{9,9'}^{\mathrm {}}$ comes close to the standard model. Including the $B\rightarrow K^*$ form-factor lattice predictions, the model comparison suggests that scenario SM+SM$'$ $(9)$ can provide an explanation of the data as efficient as in the standard model with a Bayes factor of 1:1.

A substantial reduction of uncertainties can be expected for LHCb, CMS, and ATLAS measurements of $B^0\rightarrow K^{*0}\ell ^+\ell ^-$ and $B^+\rightarrow K^+\ell ^+\ell ^-$ once they publish the analysis of their 2012 data sets. It should also be mentioned that $B\rightarrow K^*\gamma$ and $B\rightarrow K^{(*)} \ell ^+\ell ^-$ results from Belle are not based on the final reprocessed data set and that BaBar’s angular analysis of $B\rightarrow K^*\ell ^+\ell ^-$ is still preliminary. It remains to be seen whether these improved analyses further substantiate the present hints of a $1$ to $2\sigma$ deviation from the SM prediction in $\mathcal {C}_{9}^{\mathrm {}}$.

In our opinion, however, there remain two major challenges on the theory side. The first is to improve our analytic knowledge of the $1/m_b$ corrections to the exclusive decay amplitudes. The second is to reduce the uncertainty from hadronic form factors, especially at low $q^2$. Without improvements on either, there is little prospect to distinguish between small NP effects and large subleading corrections. Another point of concern are potentially large duality-violating effects that render the OPE at high $q^2$ invalid. They have been estimated, though model-dependently, to be small [7]. In this regard, the experimental verification of certain relations [8] among angular observables in $B\rightarrow K^*\ell ^+\ell ^-$ that are predicted by the OPE would be very desirable. In the case that some of these relations are not fulfilled, the analysis of the breaking pattern can provide information on duality violation but also on additional new-physics scalar and tensor interactions.